In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the study of many-body systems, namely when one looks at understanding physical properties of materials: in this case $N$ represents the number of particles, molecules, etc. of your system; the limit behavior $N\rightarrow\infty$ is of interest when this number is vast.

**My question** is very related to this concept. Say I am a bit of a theoretical physicist and I would like to ``formalize" the following procedure. Imagine you have a finite one-dimensional lattice modelled by the group $\mathbb{Z}_N$. Then, it is common (in physics) to invoke the trick $N\rightarrow\infty$ and voilà the lattice becomes infinite. Of course, this trick might be the right thing to do in certain problems, but I am presently trying to learn more about some aspects of lattice theory that depend on the (mathematical) group that describes your lattice (such as Wigner functions). For these reasons, I would like to understand **whether and how the group of integers can be obtained via limit procedure from modular groups**. In particular I would like to understand if I can construct the integers in such way using some *algebraic transformations* (such isomorphisms, homomorphisms, but not only) so that you I can make statements about group theoretical properties of the integers from what I know about modular groups using my (perhaps convoluted) limit procedure/map.

**A more concrete idea.**

I have been reading about the inverse limit procedure used in category theory and profinite completions, (I myself *do not* know category theory). I am aware that you can construct a profinite completion of the integers $\widehat{\mathbb{Z}}$ that is obtained via an inverse limit over modular groups and naturally contains $\mathbb{Z}$ (via an injective group homomorphism). This looks somewhat like what I was looking for. Still, $\widehat{\mathbb{Z}}$ is stricly larger than $\mathbb{Z}$. Still, these topics are quite alien to me and I do not know whether this construction might be something interesting to look at. I have some concrete questions about this inverse limit construction that I would like to ask here:

- Might the inverse limit procedure be a rigorous possible approach to obtain $\mathbb{Z}$ from modular groups in the lattice example I mention?
- How can I characterize which elements of $\widehat{\mathbb{Z}}$ are in $\mathbb{Z}$ (since the former is larger)? Is there a simple algorithm to do this?
- Can you use the inverse limit construction to make meaningful statements about concepts like the topology of the integers, based on your knowledge of modular groups? For instance, is it possible to induce on the integers its usual LCA topology using this limit?
**(I am particularly interested in this one)**Can you derive the**representation theory**(irreps, regular representation, etc.) of the integers (or its Pontryagin-dual group, the torus), and its**Fourier analysis**using the inverse limit procedure? (Again, from what you know about modular groups.) For instance, can you characterize the irreps of $\mathbb{Z}$ in this way.

**Both answers and references are greatly welcomed!**

topology on the space of marked groups. $\endgroup$